in PROBABILITY
ON TIME-CHANGED GAUSSIAN PROCESSES
AND THEIR ASSOCIATED FOKKER–PLANCK–
KOLMOGOROV EQUATIONS
MARJORIE G. HAHN
Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, MA 02155, USA email: marjorie.hahn@tufts.edu
KEI KOBAYASHI
Department of Mathematics, Tufts University email: kei.kobayashi@tufts.edu
JELENA RYVKINA
Department of Mathematics, Tufts University email: jelena.ryvkina@tufts.edu
SABIR UMAROV
Department of Mathematics, Tufts University email: sabir.umarov@tufts.edu
SubmittedOctober 5, 2010, accepted in final formFebruary 12, 2011 AMS 2000 Subject classification: Primary 60G15, 35Q84; Secondary 60G22.
Keywords: time-change, inverse subordinator, Gaussian process, Fokker–Planck equation, Kol-mogorov equation, fractional Brownian motion, time-dependent Hurst parameter, Volterra pro-cess.
Abstract
This paper establishes Fokker–Planck–Kolmogorov type equations for time-changed Gaussian pro-cesses. Examples include those equations for a time-changed fractional Brownian motion with dependent Hurst parameter and for a changed Ornstein–Uhlenbeck process. The time-change process considered is the inverse of either a stable subordinator or a mixture of indepen-dent stable subordinators.
1
Introduction
A one-dimensional stochastic processX = (Xt)t≥0is called aGaussian processif the random
vec-tor(Xt1, . . . ,Xtm)has a multivariate Gaussian distribution for all finite sequences 0≤ t1<· · ·<
tm<∞. The joint distributions are characterized by the mean functionE[Xt]and the covariance functionRX(s,t) =Cov(Xs,Xt). The class of Gaussian processes contains some of the most impor-tant stochastic processes in both theoretical and applied probability, including Brownian motion, fractional Brownian motion[6,8,10,32], and Volterra processes[1,9].
In this paper, Fokker–Planck–Kolmogorov type equations (FPKEs) for time-changed Gaussian pro-cesses are derived. These FPKEs are partial differential equations (PDEs) satisfied by the transi-tion probabilities of those processes. Main features of these equatransi-tions involve 1) fractransi-tional order derivatives in time and 2) a new class of operators acting on the time variable; see (14)–(16). By definition, thefractional order derivative D∗βof orderβ∈(0, 1)in the sense of Caputo–Djrbashianis given by
D∗βg(t) =Dβ∗,tg(t) = 1
Γ(1−β) Z t
0
g′(τ)
(t−τ)βdτ, (1)
withΓ(·)being Euler’s Gamma function. By convention, set D1
∗,t =d/d t. Introducing the frac-tional integration operator
Jαg(t) =Jtαg(t) = 1 Γ(α)
Z t
0
(t−τ)α−1g(τ)dτ, α >0,
one can representDβ∗,t in the formD∗β,t=Jt1−β◦(d/d t)(see[12]for details).
For the last few decades, time-fractional order FPKEs have appeared as an essential tool for the study of dynamics of various complex processes arising in anomalous diffusion in physics[26,38], finance[14,18], hydrology[5]and cell biology[31]. Using several different methods, many au-thors derive FPKEs associated with time-changed stochastic processes. For example, in[4], FPKEs for time-changed continuous Markov processes are obtained via the theory of semigroups. Paper
[15] identifies a wide class of stochastic differential equations whose associated FPKEs are rep-resented by time-fractional distributed order pseudo-differential equations. The driving processes for these stochastic differential equations are time-changed Lévy processes. Two different ap-proaches are taken, one based on the semigroup technique and the other on the time-changed Itô formula in[20]. Paper[16]provides FPKEs associated with a time-changed fractional Brownian motion from a functional-analytic viewpoint. Continuous time random walk-based approaches to derivations of time-fractional order FPKEs are illustrated in[13,25,35,36]. In the current paper, we follow the method presented in[16].
Consider a time-change processEβ= (Eβt)t≥0given by theinverse, or thefirst hitting time process,
of a β-stable subordinator Wβ = (Wtβ)t≥0 with β ∈(0, 1). The relationship between the two
processes is expressed asEβt =inf{s>0 ;Wsβ>t}. To make precise the problem being pursued in this paper, first recall the following results. For proofs of these results as well as analysis on the associated classes of stochastic differential equations, see e.g.[13,15,16,25]. Throughout the paper,all processes are assumed to start at 0.
(a) IfEβ is independent of ann-dimensional Brownian motionB, thenBand the time-changed Brownian motion(BEβ
t)have respective transition probabilitiesp(t,x)andq(t,x)satisfying
the PDEs
∂tp(t,x) = 1
2∆p(t,x) and D β
∗,tq(t,x) = 1
2∆q(t,x), (2) where∂t = ∂∂t and∆=
Pn j=1∂
2
xj =
Pn j=1
∂ ∂xj
2
, with the vector x ∈Rndenoted as x = (x1, . . . ,xn).
(b) Let L be a Lévy process whose characteristic function is given byE[ei(ξ,Lt)] =etψ(ξ) with
process(LEβ
t)have respective transition probabilitiesp(t,x)andq(t,x)satisfying the PDEs
∂tp(t,x) =L∗p(t,x) and D∗β,tq(t,x) =L∗q(t,x), (3) whereL∗is the conjugate of the pseudo-differential operator with symbolψ.
(c) IfEβ is independent of ann-dimensional fractional Brownian motionBHwith Hurst param-eterH∈(0, 1)(definition is given in Example 1), thenBH and the time-changed fractional Brownian motion(BH
Etβ
)have respective transition probabilitiesp(t,x)andq(t,x)satisfying the PDEs
∂tp(t,x) =H t2H−1∆p(t,x) and Dβ∗,tq(t,x) =H Gβ2H−1,t∆q(t,x), (4) whereGγβ,t withγ∈(−1, 1)is the operator acting ontgiven by
Gγβ,tg(t) =βΓ(γ+1)J
1−β t Ls→t−1
1
2πi
Z C+i∞
C−i∞ ˜ g(z) (sβ−zβ)γ+1dz
(t), (5)
with 0<C<sandzβ =eβLn(z), Ln(z)being the principal value of the complex logarithmic function ln(z)with cut along the negative real axis. Here ˜g(s) =L[g](s) =Lt→s[g(t)](s)
and L−1[f](t) = L−1
s→t[f(s)](t) denote the Laplace transform and the inverse Laplace transform, respectively. We use both notations in equal status, preferring the subscript nota-tion whenever the funcnota-tion to be transformed depends on more than one variable, thereby avoiding possible confusion.
Obviously (a) provides a special case of both (b) and (c). Note that in (2) and (3) the second PDE is obtained upon replacing the first order time derivative∂t in the first PDE by the fractional order derivativeDβ∗,t and the right-hand side remains unchanged. On the other hand, as (4) shows, this is not the case for a changed fractional Brownian motion; the right hand side of the time-fractional order FPKE has a different form than that of the FPKE for the untime-changed time-fractional Brownian motion. Namely, the operator G2βH−1,t instead of t2H−1 appears, which is ascribed to
dependence between increments over non-overlapping intervals of the fractional Brownian motion BH. When H = 1/2 so that the fractional Brownian motion coincides with a usual Brownian motion, the FPKEs in (4) match with those in (2). The operators{Gγβ,t;γ∈(−1, 1)}are known to satisfy the semigroup property (Proposition 3.6 in[16]).
Following the functional-analytic technique presented in[16]to obtain the time-fractional order FPKE in (4), we will establish in Theorem 3 the FPKE for the time-changed Gaussian process
(XEβ
t)under the assumption that the time-change processE
β is independent of the Gaussian pro-cessX. Moreover, generalization to time-changes which are the inverses of mixtures of indepen-dent stable subordinators will be considered in Theorem 5. In Section 4, applications of these results yield FPKEs for changed mixed fractional Brownian motions as well as those for time-changed Volterra processes. Fractional Brownian motions with time-dependent Hurst parameter H=H(t)∈(1/2, 1)are among the Volterra processes considered; see Example 3. Two equivalent forms of FPKEs for a time-changed Ornstein–Uhlenbeck process are compared in Example 5.
2
Preliminaries
(Wc tβ)t≥0= (c1/βW
β
t )t≥0in the sense of finite dimensional distributions for allc>0 (see[3,30]).
SinceWβ is strictly increasing, its inverse Eβ is continuous and nondecreasing, but no longer a Lévy process (see[23]). The density fEβ
∞with respect to the two variablestandτ.
The notion of time-change can be extended to the more general case where the time-change process is given by the inverse of an arbitrary mixture of independent stable subordinators. Let ρµ(s) =R1 represents a weighted mixture of independent stable subordinators. Similar to the identity (6), the density fEµt ofE For further properties of fEβ
t(τ)andfE
µ
t(τ), see[16,25].
The above time-change processEµis connected with thefractional derivative Dµ
∗ with distributed the FPKEs for the time-changed Brownian motion(BEµt)and the time-changed Lévy process(LE
µ
t)
are respectively given by (see[15])
D∗µ,tq(t,x) =1
2∆q(t,x) and D µ
∗,tq(t,x) =L∗q(t,x).
For investigations into PDEs with fractional derivatives with distributed orders, see[21,24,34]. The covariance function of a given zero-mean Gaussian process is symmetric and positive semi-definite; conversely, every symmetric, positive semi-definite function on [0,∞)×[0,∞)is the covariance function of some zero-mean Gaussian process (see e.g., Theorem 8.2 of [19]). Ex-amples of such functions includeRX(s,t) =s∧t for Brownian motion andRX(s,t) =σ20+s·t which is obtained from linear regression (see[29]). The sum and the product of two covariance functions for Gaussian processes are again covariance functions for some Gaussian processes. For more examples of covariance functions, consult e.g.[29]. In this paper we consider only positive definite covariance functions.
The variance functions RXj(t) =RXj(t,t)will play an important role in establishing FPKEs for
Gaussian and time-changed Gaussian processes. For differentiable variance functionsRXj(t), let
A=AX = 1 2
n
X
j=1
R′Xj(t)∂x2j. (9)
ClearlyA=1
2∆ifX is ann-dimensional Brownian motion.
Proposition 1. Let X = (X1, . . . ,Xn) be an n-dimensional zero-mean Gaussian process with co-variance functions RXj(s,t), j = 1, . . . ,n. Suppose the variance functions RXj(t) = RXj(t,t) are
differentiable on(0,∞). Then the transition probabilities p(t,x)of X satisfy the PDE
∂tp(t,x) =A p(t,x), t>0, x∈Rn, (10)
with initial condition p(0,x) =δ0(x), where A is the operator in(9)andδ0(x)is the Dirac delta function with mass on0.
Proof. Since the components Xj of X are independent zero-mean Gaussian processes, it follows that
p(t,x) =
n
Y
j=1
2πRXj(t)
−1/2
×exp
− n
X
j=1 (xj)2
2RXj(t)
(11)
fort>0. Direct computation of partial derivatives ofp(t,x)yields the equality in (10). Moreover, the initial conditionp(0,x) =δ0(x)follows immediately from the assumption that the processX starts at 0.
Remark 2. a) In Proposition 1, if the componentsXjare independent Gaussian processes with a commonvariance functionRX(t)which is differentiable, then (10) reduces to the following form which agrees with the classical FPKE:
∂tp(t,x) = 1
2R ′
X(t)∆p(t,x). (12) b) If X = (X1, . . . ,Xn) is an n-dimensional Gaussian process with mean functions m
Xj(t) and
covariance functionsRXj(s,t), and if bothmXj(t)andRXj(t) =RXj(t,t)are differentiable, then
the associated FPKE contains an additional term:
∂tp(t,x) =A p(t,x) +B p(t,x) where B=− n
X
j=1
m′Xj(t)∂xj. (13)
Such Gaussian processes include e.g. the process defined by the sum of a Brownian motion and a deterministic differentiable function.
3
FPKEs for time-changed Gaussian processes
Theorems 3 and 5 formulate the FPKE for a time-changed Gaussian process under the assumptions that the Gaussian process has positive definite covariance functions and starts at 0 while the time-change process is independent of the Gaussian process. The case where those processes are dependent is not discussed in this paper. As in Proposition 1, the variance functions play a key role here.
Theorem 3. Let X= (X1, . . . ,Xn)be an n-dimensional zero-mean Gaussian process with covariance functions RXj(s,t), j = 1, . . . ,n, and let Eβ be the inverse of a stable subordinator Wβ of index
β ∈(0, 1), independent of X . Suppose the variance functions RXj(t) =RXj(t,t) are differentiable
on(0,∞)and Laplace transformable. Then the transition probabilities q(t,x)of the time-changed Gaussian process(XEβ
t)satisfy the equivalent PDEs
Dβ∗,tq(t,x) =
n
X
j=1
Jt1−βΛβ
Xj,t∂
2
xjq(t,x), t>0, x∈R
n, (14)
∂tq(t,x) =
n
X
j=1 Λβ
Xj,t∂
2
xjq(t,x), t>0, x∈R
n, (15)
with initial condition q(0,x) =δ0(x), whereΛβ
Xj,t, j=1, . . . ,n, are the operators acting on t given
by
Λβ
Xj,tg(t) =
β 2L
−1
s→t
1
2πi
Z
C
(sβ−zβ)gRXj(sβ−zβ)˜g(z)dz
(t), (16)
with zβ =eβLn(z),Ln(z)being the principal value of the complex logarithmic functionln(z)with cut along the negative real axis, andCbeing a curve in the complex plane obtained via the transformation
ζ=zβ which leaves all the singularities ofgRXj on one side.
Proof. Letp(t,x)denote the transition probabilities of the Gaussian processX. For each x∈Rn,
it follows from the independence assumption betweenEβandX that
q(t,x) = Z∞
0
fEβ
t(τ)p(τ,x)dτ, t>0. (17)
Relationship (17) and equality (6) together yield
˜
q(s,x) =sβ−1˜p(sβ,x), s>0. (18)
Laplace transforms on both sides of (10),
where∗denotes the convolution of Laplace images and the function
g
R′
Xj(s) =sgRXj(s), s>a, (20)
exists by assumption. Equation (20) is valid sinceRXj(0) =0 due to the initial conditionXj(0) =0.
Moreover, since Eβ0 =0 with probability one, it follows thatq(0,x) =p(0,x) =δ0(x). Replacing sbysβand using the identity (18) yields
sq˜(s,x)−q(0,x) =β taking the inverse Laplace transform on both sides. Moreover, applying the fractional integral operatorJt1−βto both sides of (15) yields (14).
Remark 4. a) Representation (17) yields the estimate sup
t≥ǫ,x∈Rn|q(t,x)| ≤t≥ǫsup,x∈Rn|p(t,x)|
for anyǫ >0, which, together with the uniqueness of the solution to PDE (10) with p(0,x) =
δ0(x), guarantees uniqueness of PDE (14) (or PDE (15)) with initial conditionq(0,x) =δ0(x).
The same argument applies to the PDEs to be established in Theorem 5 as well.
b) Representation (17) together with an appropriate quadrature formula provides an approxima-tion of the soluapproxima-tion to the equivalent PDEs (14) and (15), which in turn can be used for computer based simulation of the corresponding stochastic process in a manner similar to what is done in
[2].
The next theorem extends the previous theorem to time-changes which are the inverses of mixtures of independent stable subordinators.
Theorem 5. Let X= (X1, . . . ,Xn)be an n-dimensional zero-mean Gaussian process with covariance functions RXj(s,t), j = 1, . . . ,n, and let Eµ be the inverse of a mixture Wµ of independent stable
on(0,∞)and Laplace transformable. Then the transition probabilities q(t,x)of the time-changed
the operators acting on t given by
ΛµXj,tg(t) = plane obtained via the transformationζ=ρµ(z)which leaves all the singularities ofgRXjon one side.
Proof. We only sketch the proof since it is similar to the proof of Theorem 3. Letp(t,x)denote the transition probabilities of the Gaussian processX. For each x ∈Rn, it follows from relationship
(17) with fEβ
t replaced by fE
µ
t together with equality (7) that
˜
q(s,x) = ρµ(s)
s ˜p ρµ(s),x
, s>0. (25)
Taking Laplace transforms on both sides of (10) leads to the second to last equality in (19). Letting ζ=ρµ(z)yields
yields an equation similar to (21). PDE (23) is obtained upon taking the inverse Laplace transform on both sides. Finally, applying the fractional integral operatorJt1−β and integrating with respect toµon both sides of (23) yields (22).
Remark 6. a) Ifµ=δβ0 withβ0∈(0, 1), thenΛµ
Xj,tg(t) = Λ
β0
Xj,tg(t)and the FPKEs in (22) and
(23) respectively reduce to the FPKEs in (14) and (15) withβ=β0, as expected.
b) In Theorem 5, if the componentsXjare independent Gaussian processes with acommon vari-ance functionRX(t)which is differentiable and Laplace transformable, then the FPKEs in (22) and (23) respectively reduce to the following simple forms:
D∗µ,tq(t,x) = Z1
0
J1t−βΛµX,t∆q(t,x)dµ(β); (26)
∂tq(t,x) = ΛµX,t∆q(t,x). (27)
4
Applications
This section is devoted to applications of the results established in this paper concerning FPKEs for Gaussian and time-changed Gaussian processes. For simplicity of discussion, we will consider the time-changed processEβ given by the inverse of a stable subordinatorWβ of indexβ∈(0, 1), rather than the more general time-change processEµ.
Example 1. Fractional Brownian motion. One of the most important Gaussian processes in ap-plied probability is a fractional Brownian motion BH with Hurst parameter H ∈ (0, 1). A one-dimensionalfractional Brownian motionis a zero-mean Gaussian process with covariance function
RBH(s,t) =E[BH If H = 1/2, then BH becomes a usual Brownian motion. An n-dimensional fractional Brown-ian motion is an n-dimensional process whose components are independent fractional Brownian motions with a common Hurst parameter.
Stochastic processes driven by a fractional Brownian motionBH are of increasing interest for both theorists and applied researchers due to their wide application in fields such as mathematical finance [8], solar activities[32]and turbulence[10]. The process BH, like the usual Brownian motion, has nowhere differentiable paths and stationary increments; however, it does not have independent increments. BH has the integral representation BHt = Rt
0KH(t,s)d Bs, where B is
a Brownian motion and KH(t,s)is a deterministic kernel. BH is not a semimartingale unless H=1/2, so the usual Itô’s stochastic calculus is not valid. For details of the above properties, see
[6,28].
LetBH be ann-dimensional fractional Brownian motion and letEβ be the inverse of a stable sub-ordinator of indexβ∈(0, 1), independent ofBH. Then the components ofBH share the common variance functionRBH(t) =t2H and its Laplace transformRg′
BH(s) =2HΓ(2H)/s2H. Hence,
Propo-sition 1 and Theorem 3 immediately recover both the FPKEs in (4) for the fractional Brownian motionBHand the time-changed fractional Brownian motion(BH
Eβt
). In this case,
J1t−βΛβBH,t =H G
β
2H−1,t, (29)
whereGγβ,t is the operator given in (5). Note that the curveC appearing in the expression of the
operatorΛβ
BH,t in (16) can be replaced by a vertical line{C+i r;r∈R}with 0<C<ssince the
integrand has a singularity only atz=s.
Example 2. Mixed fractional Brownian motion. Let X be an n-dimensional process defined by a finite linear combination of independent zero-mean Gaussian processes X1, . . . ,Xm: Xt =
Pm
ℓ=1aℓXℓ,t witha1, . . . ,am ∈R. For simplicity, assume that for each ℓ=1, . . . ,m, the compo-nents of the vector Xℓ = (X1
ℓ, . . . ,Xℓn)share a common variance functionRXℓ(t). Then the pro-cess X is again a Gaussian process whose components have the same variance functionRX(t) = Pm
ℓ=1a2ℓRXℓ(t). Therefore, it follows from Proposition 1 and Theorem 3 that the FPKEs forX and
(XEβ
t), under the independence assumption betweenE
β andX, are respectively given by
correspondence between the functiont2H−1and the operatorGβ2H−1,t observed in the FPKEs in (4) for the fractional Brownian motion and the time-changed fractional Brownian motion.
Amixed fractional Brownian motionis a finite linear combination of independent fractional Brow-nian motions (see[27,33]for its properties). It was introduced in[7]to discuss the price of a European call option on an asset driven by the process. The processX considered in that paper is of the formXt=Bt+a BtH, wherea∈R,Bis a Brownian motion, andB
His a fractional Brownian motion with Hurst parameterH∈(0, 1). In this situation, the FPKEs in (30), with the help of (29), yield
∂tp(t,x) = 1
2∆p(t,x) +a
2H t2H−1∆p(t,x); (31)
Dβ∗,tq(t,x) = 1
2∆q(t,x) +a
2H Gβ
2H−1,t∆q(t,x). (32)
Example 3. Fractional Brownian motion with variable Hurst parameter. Volterra processesform an important subclass of Gaussian processes. They are continuous zero-mean Gaussian processes V = (Vt)defined on a given finite interval[0,T]with integral representations of the form Vt =
Rt
0K(t,s)d Bsfor some deterministic kernelK(t,s)and Brownian motionB(see[1,9]for details).
Fractional Brownian motions are clearly an example of a Volterra process. In particular, ifBH is a fractional Brownian motion with Hurst parameterH∈(1/2, 1), thenBH =Rt
0KH(t,s)d Bswith
the kernel
KH(t,s) =cHs1/2−H
Zt
s
(r−s)H−3/2rH−1/2d r, t>s, (33)
where the positive constantcHis chosen so that the integral
Rt∧s
0 KH(t,r)KH(s,r)d rcoincides with
RBH(s,t)in (28). Increments ofBH exhibit long range dependence.
A particular interesting Volterra process is the fractional Brownian motion with time-dependent Hurst parameterH(t)suggested in Theorem 9 of[9]. Namely, supposeH(t):[0,T]→(1/2, 1)is a deterministic function satisfying the following conditions:
inf
t∈[0,T]H(t)>
1
2 and H(t)∈ S1/2+α,2 for some α∈
0, inf
t∈[0,T]H(t)−
1 2
, (34)
whereSη,2is the Sobolev-Slobodetzki space given by the closure of the spaceC1[0,T]with respect
to the semi-norm
kfk2= Z T
0 Z T
0
|f(t)−f(s)|2
|t−s|1+2η d t ds. (35)
Then representation (33) with H replaced by H(t) induces a covariance function RV(s,t) =
Rt∧s
0 KH(t)(t,r)KH(s)(s,r)d rfor some Volterra processV on[0,T]. The variance function is given
byRV(t) = t2H(t) and is necessarily continuous due to the Sobolev embedding theorem, which saysSη,2⊂C[0,T]for allη >1/2 (see e.g.[17]). Therefore,H(t)is also continuous.
Let H(t):[0,∞)→ (1/2, 1)be a differentiable function whose restriction to any finite interval
[0,T] satisfies the conditions in (34). For each T >0, let KVT(s,t)be the kernel inducing the
covariance functionRVT(s,t)of the associated Volterra process VT defined on[0,T] as above.
Hence, so is that ofRVT(s,t), which implies that the functionRX(s,t)given byRX(s,t) =RVT(s,t)
whenever 0≤ s,t ≤ T <∞ is a well-defined covariance function of a Gaussian process X on
[0,∞)whose restriction to each interval [0,T]coincides with VT. The process X represents a fractional Brownian motion with variable Hurst parameter. The variance functionRX(t) =t2H(t)is differentiable on(0,∞)by assumption and Laplace transformable due to the estimateRX(t)≤t2. Therefore, Proposition 1 and Theorem 3 can be applied to yield the FPKEs for X and the time-changed process(XEβ
t)under the independence assumption betweenE
β andX.
Example 4. Fractional Brownian motion with piecewise constant Hurst parameter. The fractional Brownian motion discussed in Example 3 has a continuously varying Hurst parameter H(t) :
[0,∞)→(1/2, 1). Here we consider a piecewise constant Hurst parameterH(t):[0,∞)→(0, 1)
which is described as
H(t) =
N
X
k=0
HkI[Tk,Tk+1)(t), (36)
where{Hk}N
k=0are constants in(0, 1),{Tk}Nk=0are fixed times such that 0=T0<T1<· · ·<TN< TN+1=∞, andI[Tk,Tk+1)denotes the indicator function over the interval[Tk,Tk+1).
For eachk=0, . . . ,N, letBHkbe ann-dimensional fractional Brownian motion with Hurst
param-eterHk. LetX be the process defined by
Xt= k−1 X
j=0 (BHj
Tj+1−B Hj
Tj) + (B
Hk
t −B Hk
Tk) whenever t∈[Tk,Tk+1). (37)
ThenX is a continuous process representing a fractional Brownian motion which involves finitely many changes of mode of Hurst parameter (described in (36)).
The transition probabilities of the processX are constructed as follows. For eachk=0, . . . ,N, let θk(t) =Hkt2Hk−1fort∈[Tk,Tk+1). Let{pk(t,x)}Nk=0be a sequence of the unique solutions to the
following initial value problems, each defined on[Tk,Tk+1)×Rn:
∂tp0(t,x) =θ0(t)∆p0(t,x), t∈(0,T1), x∈Rn, (38)
p0(0,x) =δ0(x), x∈Rn, (39)
whereδ0(x)is the Dirac delta function with mass on 0, and fork=1, . . . ,N,
∂tpk(t,x) =θk(t)∆pk(t,x), t∈(Tk,Tk+1), x∈Rn, (40)
pk(Tk,x) =pk−1(Tk−0,x), x∈Rn. (41) Define functions θ(t)and p(t,x)respectively byθ(t) =θk(t)andp(t,x) = pk(t,x)whenever t∈[Tk,Tk+1). Then the transition probabilities ofX are given byp(t,x)and satisfy
∂tp(t,x) =θ(t)∆p(t,x), t∈
SN
k=0(Tk,Tk+1), x∈Rn, (42)
p(0,x) =δ0(x), x∈Rn. (43)
problem associated with the time-changed process(XEβ
t)is given by
Dβ∗,tq(t,x) = Θ β
t ∆q(t,x), t∈(0,∞), x∈Rn, (45)
q(0,x) =δ0(x), x∈Rn, (46)
q(Tk,x) =q(Tk−0,x), x∈Rn, k=1, . . . ,N, (47)
whereΘβt is the operator acting ontdefined byΘtβ=PNk=0HkG β
2Hk−1,tI[Tk,Tk+1)(t).
Remark 7. Combining ideas in Examples 3 and 4, it is possible to construct a fractional Brownian motion having variable Hurst parameterH(t)∈(1/2, 1)with finitely many changes of mode and to establish the associated FPKEs.
Example 5. Ornstein–Uhlenbeck process. Consider the one-dimensional Ornstein–Uhlenbeck pro-cessY given by
Yt=y0e−αt+σ Z t
0
e−α(t−s)d Bs, t≥0, (48)
whereα≥0,σ >0, y0∈Rare constants and Bis a standard Brownian motion. Ifα=0, then
Yt = y0+σBt, a Brownian motion multiplied byσstarting at y0. Supposeα >0. The processY defined by (48) is the unique strong solution to the inhomogeneous linear SDE
d Yt =−αYtd t+σd Bt with Y0=y0, (49) which is associated with the SDE
dY¯t =−αY¯td E β
t +σd BEβt with ¯Y0=y0, (50)
via the dual relationships ¯Yt=YEβt andYt=
¯ YWβ
t ; see[20]for details.
Consider the zero-mean processX defined by
Xt=Yt−y0e−αt=σ
Z t
0
e−α(t−s)d Bs. (51)
X is a Gaussian process since each random variableXtis a linear transformation of the Itô stochas-tic integral of the determinisstochas-tic integrandeαs. Direct calculation yieldsRX(t) =σ2
2α(1−e
−2αt)and
f
R′X(s) = σ2
s+2α. Therefore, due to Proposition 1 and Theorem 3, the initial value problems associated withX and(XEβ
t), whereE
β is independent ofX, are respectively given by
∂tp(t,x) = σ 2
2 e −2αt∂2
x p(t,x), p(0,x) =δ0(x); (52)
Dβ∗,tq(t,x) = σ 2β
2 J
1−β t Ls→t−1
1 2πi
Z
C ∂2
x˜q(z,x) sβ−zβ+2α dz
(t), q(0,x) =δ0(x). (53)
Notice that the two processes X and(XEβ
t)are unique strong solutions to SDEs (49) and (50)
with y0 =0, respectively. Therefore, it is also possible to apply Theorem 4.1 of [15] to obtain the following forms of initial value problems which are understood in the sense of generalized functions:
∂tp(t,x) =α ∂xx p(t,x) +σ 2
2 ∂
2
x p(t,x), p(0,x) =δ0(x); (54)
D∗β,tq(t,x) =α ∂x
xq(t,x) +σ 2
2 ∂
2
xq(t,x), q(0,x) =δ0(x). (55)
Actually these FPKEs hold in the strong sense as well. For uniqueness of solutions to (54) and (55), see e.g.[11]and Corollary 3.2 of[15].
The above discussion yields the following two sets of equivalent initial value problems: (52) and (54), and (53) and (55). At first glance, PDE (52) might seem simpler or computationally more tractable than PDE (54); however, PDE (53) which is associated with the time-changed process has a more complicated form than PDE (55). A significant difference between PDEs (52) and (54) is the fact that the right-hand side of (54) can be expressed asA∗p(t,x)with thespatialoperatorA∗=
α ∂xx+σ2 2 ∂
2
x whereas the right-hand side of (52) involves both the spatial operator∂x2and the time-dependentmultiplication operator bye−2αt. This observation suggests: 1) establishing FPKEs for time-changed processes via several different forms of FPKEs for the corresponding untime-changed processes, and 2) choosing appropriate forms for handling specific problems.
Remark 8. Example 5 treated a Gaussian process given by the Itô integral of the deterministic integrand e−α(t−s). Malliavin calculus, which is valid for an arbitrary Gaussian integrator, can be regarded as an extension of Itô integration (see [19, 22,28]). It is known that Malliavin-type stochastic integrals of deterministic integrands are again Gaussian processes. Therefore, if the variance function of such a stochastic integral satisfies the technical conditions specified in Theorem 3, then the FPKE for the time-changed stochastic integral is explicitly given by (14), or equivalently, (15).
Acknowledgements
The authors thank an anonymous referee for helpful comments and suggestions.
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